Why is the energy for an electron-hole pair for $\rm CdTe$ bigger than the band gap energy? Let's take the indirect semiconductors Si, Ge & Diamond. All these semiconductors are indirect, meaning that the maximum of the valence band is not directly under the minimum of the conduction band. This is how we can explain that the band gap for Si is $1.12$ eV, whereas the average required energy to cerate an electron-hole pair is $3.65$ eV (basically, the energy difference goes into phonons, heat).
But then, for the direct semiconductor $\rm CdTe$ (Cadmium-Tellur), why is the required energy for the creation of an $e/h$ pair $4.43$ eV, whereas the band gap is $1.44$ eV!? Very similar numbers also hold for GaAs (reference: Kolanoski-Wermes "Particle Detectors. Fundamentals and Applications. 2020. p. 261").
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From the text:

I thought that for direct semiconductors, we don't have phonon excitations.. But okay, I was probably wrong here.
 A: Here is an image of the photoluminescence from bulk CdTe layers in a heterostructure device take from this reference https://doi.org/10.1063/1.4803911. Hopefully we can agree that the energy of the emitted light is in the range 1.4-1.5eV, which agrees with your band gap value.
So we can also say that electron holes pair exist in the material with this energy difference, therefore there is some non-zero oscillator strength coupling the electron and hole states, therefore the material will also absorb and emit at this energy. In fact that’s a thermodynamic requirement via Kirchoff’s law.
I think you are reading about using CdTe as a particle detector, this will have certain constraints and limits of applicability, and may explain why the higher value is used as a figure of merit. But if we characterise the band structure, it absorbs and emits fine around 1.45eV
It could also be that the 4.4eV in the table is the electron affinity: the energy to promote an electron from the conduction band to the vacuum level. Which I guess would make sense if using the photoelectric effect to detect particles.

A: The exciton creation energy should be the direct band gap energy minus the binding energy, which is small. Here the value of 1.6 eV is given for the exciton energy in CdTe: https://www.nextnano.com/manual/nextnano3/tutorials/exciton_1D.html
A: The band gap is the minimum amount of energy needed to produce an e/h pair (1.44 eV in CdTe). But if you input 1.44 eV the energy can also be used to
heat up the lattice. Therefore, on AVERAGE 4.43 eV are consumed per produced e/h pair, because from one event to the next different amounts of energy are dissipated into e/h creation and elsewhere, most notably into lattice oscillations (phonon creation). This, to first order, has nothing to do with whether a material is an indirect or direct semiconductor. The latter means that in addition to energy transfer also (crystal) momentum transfer is required for a transition from valence to conduction band.
